Which distribution distances are sublinearly testable?
© Copyright 2018 by SIAM. Given samples from an unknown distribution p and a description of a distribution q, are p and q close or far? This question of "identity testing" has received significant attention in the case of testing whether p and q are equal or far in total variation distance...
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| Format: | Article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2022
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| Online Access: | https://hdl.handle.net/1721.1/143461 |
| _version_ | 1826193064858746880 |
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| author | Daskalakis, C Kamath, G Wright, J |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Daskalakis, C Kamath, G Wright, J |
| author_sort | Daskalakis, C |
| collection | MIT |
| description | © Copyright 2018 by SIAM. Given samples from an unknown distribution p and a description of a distribution q, are p and q close or far? This question of "identity testing" has received significant attention in the case of testing whether p and q are equal or far in total variation distance. However, in recent work [VV11a, ADK15, DP17], the following questions have been been critical to solving problems at the frontiers of distribution testing: Alternative Distances: Can we test whether p and q are far in other distances, say Hellinger? Tolerance: Can we test when p and q are close, rather than equal? And if so, close in which distances? Motivated by these questions, we characterize the complexity of distribution testing under a variety of distances, including total variation, '2, Hellinger, Kullback-Leibler, and 2. For each pair of distances d1 and d2, we study the complexity of testing if p and q are close in d1 versus far in d2, with a focus on identifying which problems allow strongly sublinear testers (i.e., those with complexity O(n1) for some > 0 where n is the size of the support of the distributions p and q). We provide matching upper and lower bounds for each case. We also study these questions in the case where we only have samples from q (equivalence testing), showing qualitative differences from identity testing in terms of when tolerance can be achieved. Our algorithms fall into the classical paradigm of ℓ2-statistics, but require crucial changes to handle the challenges introduced by each distance we consider. Finally, we survey other recent results in an attempt to serve as a reference for the complexity of various distribution testing problems. |
| first_indexed | 2024-09-23T09:33:12Z |
| format | Article |
| id | mit-1721.1/143461 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:33:12Z |
| publishDate | 2022 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/1434612023-04-20T14:55:20Z Which distribution distances are sublinearly testable? Daskalakis, C Kamath, G Wright, J Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Plasma Science and Fusion Center © Copyright 2018 by SIAM. Given samples from an unknown distribution p and a description of a distribution q, are p and q close or far? This question of "identity testing" has received significant attention in the case of testing whether p and q are equal or far in total variation distance. However, in recent work [VV11a, ADK15, DP17], the following questions have been been critical to solving problems at the frontiers of distribution testing: Alternative Distances: Can we test whether p and q are far in other distances, say Hellinger? Tolerance: Can we test when p and q are close, rather than equal? And if so, close in which distances? Motivated by these questions, we characterize the complexity of distribution testing under a variety of distances, including total variation, '2, Hellinger, Kullback-Leibler, and 2. For each pair of distances d1 and d2, we study the complexity of testing if p and q are close in d1 versus far in d2, with a focus on identifying which problems allow strongly sublinear testers (i.e., those with complexity O(n1) for some > 0 where n is the size of the support of the distributions p and q). We provide matching upper and lower bounds for each case. We also study these questions in the case where we only have samples from q (equivalence testing), showing qualitative differences from identity testing in terms of when tolerance can be achieved. Our algorithms fall into the classical paradigm of ℓ2-statistics, but require crucial changes to handle the challenges introduced by each distance we consider. Finally, we survey other recent results in an attempt to serve as a reference for the complexity of various distribution testing problems. 2022-06-17T14:32:34Z 2022-06-17T14:32:34Z 2018-01-01 2022-06-17T14:24:16Z Article http://purl.org/eprint/type/ConferencePaper https://hdl.handle.net/1721.1/143461 Daskalakis, C, Kamath, G and Wright, J. 2018. "Which distribution distances are sublinearly testable?." Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. en 10.1137/1.9781611975031.175 Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial and Applied Mathematics SIAM |
| spellingShingle | Daskalakis, C Kamath, G Wright, J Which distribution distances are sublinearly testable? |
| title | Which distribution distances are sublinearly testable? |
| title_full | Which distribution distances are sublinearly testable? |
| title_fullStr | Which distribution distances are sublinearly testable? |
| title_full_unstemmed | Which distribution distances are sublinearly testable? |
| title_short | Which distribution distances are sublinearly testable? |
| title_sort | which distribution distances are sublinearly testable |
| url | https://hdl.handle.net/1721.1/143461 |
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